Galois structure of homogeneous coordinate rings

Details Galois structure of homogeneous coordinate rings Galois structure of homogeneous coordinate rings

2005-04-13

arxiv.org/abs/math/0504281v3

Trans. Amer. Math. Soc. 360 (2008), no. 12, 6269-6301.

Mathematics

Group Theory

27 pages. The abstract and introduction have been changed; the article has been shortened

Scientific paper

10.1090/S0002-9947-08-04436-X

Suppose $G$ is a finite group acting on a projective scheme $X$ over a commutative Noetherian ring $R$. We study the $RG$-modules $\HH^0(X,\mathcal{F} \otimes \mathcal{L}^n)$ when $n \ge 0$, and $\mathcal{F}$ and $\mathcal{L}$ are coherent $G$-sheaves on $X$ such that $\mathcal{L}$ is an ample line bundle. We show that the classes of these modules in the Grothendieck group $G_0(RG)$ of all finitely generated $RG$-modules lie in a finitely generated subgroup. Under various hypotheses, we show that there is a finite set of indecomposable $RG$-modules such that each $\HH^0(X,\mathcal{F} \otimes \mathcal{L}^n)$ is a direct sum of these indecomposables, with multiplicites given by generalized Hilbert polynomials for $n >> 0$.

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